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Register a new userGraph distances in scale-free percolation: the logarithmic case
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Scale-free percolation is a spatial random graph model with vertex set $mathbb{Z}^d$. Vertices $x$ and $y$ are connected with probability depending on i.i.d. vertex weights and the Euclidean distance. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances (in comparison to Euclidean distances). Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.
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The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $alpha$ controls the tail distribution of the major axis distribution and we focus on the regime $alpha in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between points $x$ and $y$ behaves roughly as $c loglog |x-y|$.
We study the critical behavior for percolation on inhomogeneous random networks on $n$ vertices, where the weights of the vertices follow a power-law distribution with exponent $tau in (2,3)$. Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as $ntoinfty$. We identify the critical window for a host of scale-free random graph models such as the Norros-Reittu model, Chung-Lu model and generalized random graphs. Surprisingly, there exists a finite time inside the critical window, after which, we see a sudden emergence of a tiny giant component. This is a novel behavior which is in contrast with the critical behavior in other known universality classes with $tau in (3,4)$ and $tau >4$. Precisely, for edge-retention probabilities $pi_n = lambda n^{-(3-tau)/2}$, there is an explicitly computable $lambda_c>0$ such that the critical window is of the form $lambda in (0,lambda_c),$ where the largest clusters have size of order $n^{beta}$ with $beta=(tau^2-4tau+5)/[2(tau-1)]in[sqrt{2}-1, tfrac{1}{2})$ and have non-degenerate scaling limits, while in the supercritical regime $lambda > lambda_c$, a unique `tiny giant component of size $sqrt{n}$ emerges. For $lambda in (0,lambda_c),$ the scaling limit of the maximum component sizes can be described in terms of components of a one-dimensional inhomogeneous percolation model on $mathbb{Z}_+$ studied in a seminal work by Durrett and Kesten. For $lambda>lambda_c$, we prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight $Omega(sqrt{n})$.
Recent studies introduced biased (degree-dependent) edge percolation as a model for failures in real-life systems. In this work, such process is applied to networks consisting of two types of nodes with edges running only between nodes of unlike type. Such bipartite graphs appear in many social networks, for instance in affiliation networks and in sexual contact networks in which both types of nodes show the scale-free characteristic for the degree distribution. During the depreciation process, an edge between nodes with degrees k and q is retained with probability proportional to (kq)^(-alpha), where alpha is positive so that links between hubs are more prone to failure. The removal process is studied analytically by introducing a generating functions theory. We deduce exact self-consistent equations describing the system at a macroscopic level and discuss the percolation transition. Critical exponents are obtained by exploiting the Fortuin-Kasteleyn construction which provides a link between our model and a limit of the Potts model.
Biased (degree-dependent) percolation was recently shown to provide new strategies for turning robust networks fragile and vice versa. Here we present more detailed results for biased edge percolation on scale-free networks. We assume a network in which the probability for an edge between nodes $i$ and $j$ to be retained is proportional to $(k_ik_j)^{-alpha}$ with $k_i$ and $k_j$ the degrees of the nodes. We discuss two methods of network reconstruction, sequential and simultaneous, and investigate their properties by analytical and numerical means. The system is examined away from the percolation transition, where the size of the giant cluster is obtained, and close to the transition, where nonuniversal critical exponents are extracted using the generating functions method. The theory is found to agree quite well with simulations. By introducing an extension of the Fortuin-Kasteleyn construction, we find that biased percolation is well described by the $qto 1$ limit of the $q$-state Potts model with inhomogeneous couplings.
For massless vertex-transitive transient graphs, the percolation phase transition for the level sets of the Gaussian free field on the associated continuous cable system is particularly well understood, and in particular the associated critical parameter $widetilde{h}_*$ is always equal to zero. On general transient graphs, two weak conditions on the graph $mathcal{G}$ are given in arXiv:2101.05800, each of which implies one of the two inequalities $widetilde{h}_*leq0$ and $widetilde{h}_*geq0.$ In this article, we give two counterexamples to show that none of these two conditions are necessary, prove that the strict inequality $widetilde{h}_*<0$ is typical on massive graphs with bounded weights, and provide an example of a graph on which $widetilde{h}_*=infty.$ On the way, we obtain another characterization of random interlacements on massive graphs, as well as an isomorphism between the Gaussian free field and the Doob $mathit{mathbf{h}}$-transform of random interlacements, and between the two-dimensional pinned free field and random interlacements.
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